After having read abstract concepts of algebraic curves, I have trouble dealing with actual examples. For instance, why is the ϕ=yx a rational function on the curve F=y2+y+x2? I know that any rational function on this curve should be of the form {ϕ=fg:f,g∈K[x,y](F),g≠0}, but what do I need to actually check to show that this is a rational function on F? Any help will be good

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After having read abstract concepts of algebraic curves, I have trouble dealing with actual examples. For instance, why is the ϕ=yx a rational function on the curve F=y2+y+x2? I know that any rational function on this curve should be of the form {ϕ=fg:f,g∈K[x,y](F),g≠0}, but what do I need to actually check to show that this is a rational function on F?  Any help will be good

copausc20 · Accepted Answer

Your definition of a rational function is just fine for where you&#039;re at and your function exactly fits it. To see this, note that you need to verify that f=y and g=x are elements of K[x,y](F) and that g≠0 (as a function). But this is clear.

After having read abstract concepts of algebraic curves, I have trouble dealing with actual examples....

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